返回题库

HMMT 十一月 2015 · GEN 赛 · 第 4 题

HMMT November 2015 — GEN Round — Problem 4

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. Chords AB and CD of a circle are perpendicular and intersect at a point P . If AP = 6 , BP = 12, and
    CD = 22, find the area of the circle.
解析

英文解析

  1. Chords AB and CD of a circle are perpendicular and intersect at a point P . If AP = 6 , BP = 12, and
    CD = 22, find the area of the circle.
    Proposed by: Sam Korsky
    Answer: 130 π
    Let O be the center of the circle and let M be the midpoint of segment AB and let N be the midpointof segment CD . Since quadrilateral OM P N is a rectangle we have that ON = M P = AM − AP = 3
    √so
    √ √
    2 2
    OC = ON + N C = 9 + 121 = 130
    Hence the desired area is 130 π .